Mechanics of Irregular Honeycomb Structures

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1 Mechanics of Irregular Honeycomb Structures S. Adhikari 1, T. Mukhopadhyay 1 Chair of Aerospace Engineering, College of Engineering, Swansea University, Singleton Park, Swansea SA2 8PP, UK Sixth International Congress on Computational Design Optimization and Simulation, 216, IIT Bombay, India Adhikari (Swansea) Mechanics of irregular honeycomb structures 28 June 216 1

2 Outline 1 Introduction Regular Honeycomb 2 Equivalent elastic properties of random irregular honeycombs Longitudinal Young s modulus (E 1 ) Transverse Young s modulus (E 2 ) Poisson s ratio ν 12 Poisson s ratio ν 21 Shear modulus (G 12 ) 3 Uncertainty modelling and simulation 4 Results and discussions Numerical results for the homogenised in-plane properties Main observations 5 Conclusions Adhikari (Swansea) Mechanics of irregular honeycomb structures 28 June 216 2

3 Swansea University Adhikari (Swansea) Mechanics of irregular honeycomb structures 28 June 216 3

4 Swansea University Adhikari (Swansea) Mechanics of irregular honeycomb structures 28 June 216 4

5 Introduction Lattice structures Lattice structures are abundant in man-made and natural systems at various length scales Lattice structures are made of periodic identical/near-identical geometric units Among various lattice geometries (triangle, square, rectangle, pentagon, hexagon), hexagonal lattice is most common (note that hexagon is the highest space filling pattern in 2D). This talk is about in-plane elastic properties of 2D hexagonal lattice structures - commonly known as honeycombs Adhikari (Swansea) Mechanics of irregular honeycomb structures 28 June 216 5

6 Lattice structures - nano scale Introduction Single layer graphene sheet and born nitride nano sheet Adhikari (Swansea) Mechanics of irregular honeycomb structures 28 June 216 6

7 Introduction Lattice structures - nature Top left: cork, top right: balsa, next down left: sponge, next down right: trabecular bone, next down left: coral, next down right: cuttlefish bone, bottom left: leaf tissue, bottom right: plant stem, third column - epidermal cells (from web.mit.edu) Adhikari (Swansea) Mechanics of irregular honeycomb structures 28 June 216 7

8 Introduction Lattice structures - man made (a) Automotive: BMW i3 (c) Civil engineering: building frame Adhikari (Swansea) (b) Aerospace carbon fibre (d) Architecture Mechanics of irregular honeycomb structures 28 June 216 8

9 Introduction Some questions of general interest Shall we consider lattices as structures or materials from a mechanics point of view? At what relative length-scale a lattice structure can be considered as a material with equivalent elastic properties? In what ways structural irregularities mess up equivalent elastic properties? Can we evaluate it in a quantitative as well as in a qualitative manner? What is the consequence of random structural irregularities on the homogenisation approach in general? Can we obtain statistical measures? Is there any underlying ergodic behaviour for large random lattices so that ensemble statistics is close to a sample statistics? How large is large? How can we efficiently compute equivalent elastic properties of random lattice structures? Adhikari (Swansea) Mechanics of irregular honeycomb structures 28 June 216 9

10 Introduction Regular lattice structures Honeycombs have been modelled as a continuous solid with an equivalent elastic moduli throughout its domain. This approach eliminates the need of detail finite element modelling of honeycombs in complex structural systems like sandwich structures. Extensive amount of research has been carried out to predict the equivalent elastic properties of regular honeycombs consisting of perfectly periodic hexagonal cells (El-Sayed et al., 1979; Gibson and Ashby, 1999; Goswami, 26; Masters and Evans, 1996; Zhang and Ashby, 1992). Analysis of two dimensional honeycombs dealing with in-plane elastic properties are commonly based on unit cell approach, which is applicable only for perfectly periodic cellular structures. Adhikari (Swansea) Mechanics of irregular honeycomb structures 28 June 216 1

11 Introduction Regular Honeycomb Equivalent elastic properties of regular honeycombs Unit cell approach - Gibson and Ashby (1999) (e) Regular hexagon (θ = 3 ) (f) Unit cell We are interested in homogenised equivalent in-plane elastic properties This way, we can avoid a detailed structural analysis considering all the beams and treat it as a material Adhikari (Swansea) Mechanics of irregular honeycomb structures 28 June

12 Introduction Regular Honeycomb Equivalent elastic properties of regular honeycombs The cell walls are treated as beams of thickness t, depth b and Young s modulus E s. l and h are the lengths of inclined cell walls having inclination angle θ and the vertical cell walls respectively. The equivalent elastic properties are: ( ) 3 t cos θ E 1 = E s l ( h l + sin θ) sin 2 (1) θ ( ) 3 t ( h l + sin θ) E 2 = E s l cos 3 (2) θ cos 2 θ ν 12 = ( h l + sin θ) sin θ ν 21 = ( h l + sin θ) sin θ cos 2 (4) θ ( ) 3 ( t h l + sin θ ) G 12 = E s l ( h ) 2 (5) l (1 + 2 h l ) cos θ (3) Adhikari (Swansea) Mechanics of irregular honeycomb structures 28 June

13 Introduction Regular Honeycomb Finite element modelling and verification A finite element code has been developed to obtain the in-plane elastic moduli numerically for honeycombs. Each cell wall has been modelled as an Euler-Bernoulli beam element having three degrees of freedom at each node. For E 1 and ν 12 : two opposite edges parallel to direction-2 of the entire honeycomb structure are considered. Along one of these two edges, uniform stress parallel to direction-1 is applied while the opposite edge is restrained against translation in direction-1. Remaining two edges (parallel to direction-1) are kept free. For E 2 and ν 21 : two opposite edges parallel to direction-1 of the entire honeycomb structure are considered. Along one of these two edges, uniform stress parallel to direction-2 is applied while the opposite edge is restrained against translation in direction-2. Remaining two edges (parallel to direction-2) are kept free. For G 12 : uniform shear stress is applied along one edge keeping the opposite edge restrained against translation in direction-1 and 2, while the remaining two edges are kept free. Adhikari (Swansea) Mechanics of irregular honeycomb structures 28 June

14 Introduction Regular Honeycomb Finite element modelling and verification Ratio of elastic modulus E 1 E 2 ν 12 ν 21 G Number of unit cells θ = 3, h/l = 1.5. FE results converge to analytical predictions after 1681 cells. Adhikari (Swansea) Mechanics of irregular honeycomb structures 28 June

15 Equivalent elastic properties of random irregular honeycombs Irregular lattice structures (g) Cedar wood (h) Manufactured honeycomb core (i) Graphene image (j) Fabricated CNT surface Adhikari (Swansea) Mechanics of irregular honeycomb structures 28 June

16 Equivalent elastic properties of random irregular honeycombs Irregular lattice structures A significant limitation of the aforementioned unit cell approach is that it cannot account for the spatial irregularity, which is practically inevitable. Spatial irregularity in honeycomb may occur due to manufacturing uncertainty, structural defects, variation in temperature, pre-stressing and micro-structural variability in honeycombs. To include the effect of irregularity, voronoi honeycombs have been considered in several studies (Li et al., 25; Zhu et al., 21, 26). The effect of different forms of irregularity on elastic properties and structural responses of honeycombs are generally based on direct finite element (FE) simulation. In the FE approach, a small change in geometry of a single cell may require completely new geometry and meshing of the entire structure. In general this makes the entire process time-consuming and tedious. The problem becomes even worse for uncertainty quantification of the responses associated with irregular honeycombs, where the expensive finite element model is needed to be simulated for a large number of samples while using a Monte Carlo based approach. Adhikari (Swansea) Mechanics of irregular honeycomb structures 28 June

17 Equivalent elastic properties of random irregular honeycombs Irregular lattice structures Direct numerical simulation to deal with irregularity in honeycombs may not necessarily provide proper understanding of the underlying physics of the system. An analytical approach could be a simple, insightful, yet an efficient way to obtain effective elastic properties of honeycombs. This work develops a structural mechanics based analytical framework for predicting equivalent in-plane elastic properties of irregular honeycomb having spatially random variations in cell angles. Closed-form analytical expressions will be derived for equivalent in-plane elastic properties. Adhikari (Swansea) Mechanics of irregular honeycomb structures 28 June

18 Equivalent elastic properties of random irregular honeycombs The philosophy of the analytical approach for irregular honeycombs The key idea to obtain the effective in-plane elastic moduli of the entire irregular honeycomb structure is that it is considered to be consisted of several Representative Unit Cell Elements (RUCE) having different individual elastic moduli. The expressions for elastic moduli of a RUCE is derived first and subsequently the expressions for effective in-plane elastic moduli of the entire irregular honeycomb are derived by assembling the individual elastic moduli of these RUCEs using basic principles of mechanics (divide and concur!). Adhikari (Swansea) Mechanics of irregular honeycomb structures 28 June

19 Equivalent elastic properties of random irregular honeycombs Longitudinal Young s modulus (E 1 ) Longitudinal Young s modulus (E 1 ) To derive the expression of longitudinal Young s modulus for a RUCE (E 1U ), stress σ 1 is applied in direction-1 as shown below: (k) (l) Figure: RUCE and free-body diagram used in the analysis for E 1 Adhikari (Swansea) Mechanics of irregular honeycomb structures 28 June

20 Equivalent elastic properties of random irregular honeycombs Longitudinal Young s modulus (E 1 ) Elastic property of a representative unit cell element (RUCE) The inclined cell walls having inclination angle α and β do not have any contribution in the analysis, as the stresses applied on them in two opposite directions neutralise each other. The remaining structure except these two inclined cell walls is symmetric. The applied stresses cause the inclined cell walls having inclination angle θ to bend. From the condition of equilibrium, the vertical forces C in the free-body diagram of these cell walls need to be zero. The cell walls are treated as beams of thickness t, depth b and Young s modulus E s. l and h are the lengths of inclined cell walls having inclination angle θ and the vertical cell walls respectively. Therefore, we have where M = Pl sin θ 2 (6) P = σ 1 (h + l sin θ)b (7) Adhikari (Swansea) Mechanics of irregular honeycomb structures 28 June 216 2

21 Equivalent elastic properties of random irregular honeycombs Longitudinal Young s modulus (E 1 ) Elastic property of a representative unit cell element (RUCE) From the standard beam theory, the deflection of one end compared to the other end of the cell wall can be expressed as δ = Pl3 sin θ 12E s I (8) where I is the second moment of inertia of the cell wall, that is I = bt 3 /12. The component of δ parallel to direction-1 is δ sin θ. The strain parallel to direction-1 becomes ɛ 1 = δ sin θ (9) l cos θ Thus the Young s modulus in direction-1 for a RUCE can be expressed as E 1U = σ 1 ɛ 1 = E s ( ) 3 t cos θ ( ) l h + sin θ sin 2 θ l (1) Next we use this to obtain E 1 for the entire structure. Adhikari (Swansea) Mechanics of irregular honeycomb structures 28 June

22 Equivalent elastic properties of random irregular honeycombs Longitudinal Young s modulus (E 1 ) Elastic property of the entire irregular honeycomb (b) Idealized j th strip (a) Entire idealized irregular honeycomb structure (c) Idealized i th cell in j th strip Figure: Free-body diagrams of idealized irregular honeycomb structure in the proposed analysis of E 1 The entire irregular honeycomb structure is assumed to have m and n number of RUCEs in direction-1 and direction-2 respectively. A particular cell having position at i th column and j th row is represented as (i,j), where i = 1, 2,..., m and j = 1, 2,..., n. Adhikari (Swansea) Mechanics of irregular honeycomb structures 28 June

23 Equivalent elastic properties of random irregular honeycombs Longitudinal Young s modulus (E 1 ) Elastic property of the entire irregular honeycomb To obtain E 1eq, stress σ 1 is applied in direction-1. If the deformation compatibility condition of j th strip (as highlighted in the figure) is considered, the total deformation due to stress σ 1 of that particular strip ( 1 ) is the summation of individual deformations of each RUCEs in direction-1, while deformation of each of these RUCEs in direction-2 is the same. Thus for the j th strip 1 = The equation (11) can be rewritten as m 1ij (11) i=1 ɛ 1 L = m ɛ 1ij L ij (12) i=1 where ɛ 1 and L represent strain and dimension in direction-1 of respective elements. Adhikari (Swansea) Mechanics of irregular honeycomb structures 28 June

24 Equivalent elastic properties of random irregular honeycombs Longitudinal Young s modulus (E 1 ) Elastic property of the entire irregular honeycomb Equation (12) leads to σ 1 L = Eˆ 1j m i=1 σ 1 L ij E 1Uij (13) From equation (13), equivalent Young s modulus of j th strip ( E ˆ 1j ) can be expressed as m ˆ E 1j = i=1 m i=1 l ij cos θ ij l ij cos θ ij E 1Uij where θ ij is the inclination angle of the cell walls having length l ij in the RUCE positioned at (i,j). (14) Adhikari (Swansea) Mechanics of irregular honeycomb structures 28 June

25 Equivalent elastic properties of random irregular honeycombs Longitudinal Young s modulus (E 1 ) Elastic property of the entire irregular honeycomb After obtaining the Young s moduli of n number of strips, they are assembled to achieve the equivalent Young s modulus of the entire irregular honeycomb structure (E 1eq ) using force equilibrium and deformation compatibility conditions. σ 1 Bb = n σ 1j B j b (15) j=1 where B j is the dimension of j th strip in direction-2 and B = n B j. b represents the depth of honeycomb. As strains in direction-1 for each of the n strips are same to satisfy the deformation compatibility condition, equation (15) leads to n n E 1eq = Eˆ 1j B j (16) B j j=1 j=1 j=1 Adhikari (Swansea) Mechanics of irregular honeycomb structures 28 June

26 Equivalent elastic properties of random irregular honeycombs Longitudinal Young s modulus (E 1 ) Elastic property of the entire irregular honeycomb Using equation (14) and equation (16), equivalent Young s modulus in direction-1 of the entire irregular honeycomb structure (E 1eq ) can be expressed as: Equivalent E 1 E 1eq = 1 n B j j=1 m n l ij cos θ ij i=1 m l ij cos θ ij B j (17) j=1 E 1Uij i=1 Here Young s modulus in direction-1 of a RUCE positioned at (i,j) is E 1Uij, which can be obtained from equation (1) as ( ) 3 t cos θ E 1Uij = E s ( ) ij (18) l ij h + sin θ ij sin 2 θ ij l ij Adhikari (Swansea) Mechanics of irregular honeycomb structures 28 June

27 Equivalent elastic properties of random irregular honeycombs Transverse Young s modulus (E 2 ) Transverse Young s modulus (E 2 ) To derive the expression of transverse Young s modulus for a RUCE (E 2U ), stress σ 2 is applied in direction-2 as shown below: (a) (b) (c) Figure: RUCE and free-body diagram used in the proposed analysis for E 2 Adhikari (Swansea) Mechanics of irregular honeycomb structures 28 June

28 Equivalent elastic properties of random irregular honeycombs Transverse Young s modulus (E 2 ) Elastic property of a representative unit cell element (RUCE) Total deformation of the RUCE in direction-2 consists of three components, namely deformation of the cell wall having inclination angle α, deformation of the cell walls having inclination angle θ and deformation of the cell wall having inclination angle β. If the remaining structure except the two inclined cell walls having inclination angle α and β is considered, two forces that act at joint B are W and M 1. For the cell wall having inclination angle α, effect of the bending moment M 1 generated due to application of W at point D is only to create rotation (φ) at the joint B. Vertical deformation of the cell wall having inclination angle α has two components, bending deformation in direction-2 and rotational deformation due the rotation of joint B. Adhikari (Swansea) Mechanics of irregular honeycomb structures 28 June

29 Equivalent elastic properties of random irregular honeycombs Transverse Young s modulus (E 2 ) Elastic property of a representative unit cell element (RUCE) After some algebra and mechanics, the total deformation in direction-2 of the RUCE due to the application of stresses σ 2 is δ v = σ 2l cos θ E s t ( 3 ( cos 2l 3 cos 2 θ + 8s 3 2 ) ) α sin 3 α + cos2 β sin 3 + 2s 2 l(cot 2 α + cot 2 β) β (19) The strain in direction-2 can be obtained as ɛ 2 = δ v h + 2s + 2l sin θ (2) Adhikari (Swansea) Mechanics of irregular honeycomb structures 28 June

30 Equivalent elastic properties of random irregular honeycombs Transverse Young s modulus (E 2 ) Elastic property of a representative unit cell element (RUCE) Therefore, the Young s modulus in direction-2 of a RUCE can be expressed as E 2U = σ 2 ɛ 2 = E s ( ) 3 t l ( ( s cos θ 2 cos 2 θ + 8 l ( h + 2 s l l ) 3 ( cos 2 α sin 3 α + cos2 β sin 3 β ) + 2 sin θ ) ( s ) (cot 2 α + cot β)) 2 l (21) Adhikari (Swansea) Mechanics of irregular honeycomb structures 28 June 216 3

31 Equivalent elastic properties of random irregular honeycombs Transverse Young s modulus (E 2 ) Elastic property of the entire irregular honeycomb To derive the expression of equivalent Young s modulus in direction-2 for the entire irregular honeycomb structure (E 2eq ), the Young s moduli for the constituting RUCEs (E 2U ) are assembled. (b) Idealized j th strip (a) Entire idealized irregular honeycomb structure (c) Idealized i th cell in j th strip Figure: Free-body diagrams of idealized irregular honeycomb structure Adhikari (Swansea) Mechanics of irregular honeycomb structures 28 June

32 Equivalent elastic properties of random irregular honeycombs Transverse Young s modulus (E 2 ) Elastic property of the entire irregular honeycomb When the force equilibrium under the application of stress σ 2 of j th strip (as highlighted in 4(b)) is considered: ( m ) ( m ) σ 2 2l ij cos θ ij b = σ 2ij 2l ij cos θ ij b (22) i=1 By deformation compatibility condition, strains of each RUCE in direction-2 of the j th strip are same. Equation (22), rewritten as ( m ) ( m ) Eˆ 2j l ij cos θ ij ɛ = E 2Uij l ij cos θ ij ɛ ij i=1 where ɛ ij = ɛ, for i = 1, 2...m in the j th strip. Eˆ 2j is the equivalent elastic modulus in direction-2 of the j th strip: m ˆ E 2j = i=1 i=1 i=1 E 2Uij l ij cos θ ij (23) (24) m l ij cos θ ij i=1 Adhikari (Swansea) Mechanics of irregular honeycomb structures 28 June

33 Equivalent elastic properties of random irregular honeycombs Transverse Young s modulus (E 2 ) Elastic property of the entire irregular honeycomb Total deformation of the entire honeycomb in direction-2 ( 2 ) is the sum of deformations of each strips in that direction, n 2 = 2ij (25) Equation (25) can be rewritten as ɛ 2 B = j=1 n ɛ 2j B j (26) where ɛ 2, ɛ 2j and B j represent total strain of the entire honeycomb structure in direction-2, strain of j th strip in direction-2 and dimension in direction-2 of j th strip respectively. From equation (26) we have n σ 2 B j j=1 E 2eq = j=1 n j=1 σ 2 B j ˆ E 2j (27) Adhikari (Swansea) Mechanics of irregular honeycomb structures 28 June

34 Equivalent elastic properties of random irregular honeycombs Transverse Young s modulus (E 2 ) Elastic property of the entire irregular honeycomb From equation (24) and equation (27), the Young s modulus in direction-2 of the entire irregular honeycomb structure can be expressed as Equivalent E 2 E 2eq = n B j j=1 1 m l ij cos θ ij i=1 m E 2Uij l ij cos θ ij i=1 n B j (28) j=1 Here Young s modulus in direction-2 of a RUCE positioned at (i,j) is E 2Uij, which can be obtained from equation (21). Adhikari (Swansea) Mechanics of irregular honeycomb structures 28 June

35 Equivalent elastic properties of random irregular honeycombs Transverse Young s modulus (E 2 ) Special case: classical deterministic results The expressions of Young s moduli for randomly irregular honeycombs (equation (17) and (28)) reduces to the formulae provided by Gibson and Ashby (Gibson and Ashby, 1999) in case of uniform honeycombs (i.e. B 1 = B 2 =... = B n ; s = h/2; α = β = 9 ; l ij = l and θ ij = θ, for all i and j). By applying the conditions B 1 = B 2 =... = B n ; l ij = l and θ ij = θ, equation (17) and (28) reduce to E 1U and E 2U respectively. For s = h/2 and α = β = 9, E 1U and E 2U produce the same expressions for Young s moduli of uniform honeycomb as presented by Gibson and Ashby (Gibson and Ashby, 1999). In the case of regular uniform honeycombs (θ = 3 ) E1 = E ( ) 3 2 t = 2.3 (29) E s E s l where E1 and E 2 denote the Young moduli of uniform regular honeycombs in longitudinal and transverse direction respectively. Adhikari (Swansea) Mechanics of irregular honeycomb structures 28 June

36 Equivalent elastic properties of random irregular honeycombs Poisson s ratio ν 12 Poisson s ratio ratio ν 12 Poisson s ratios are calculated by taking the negative ratio of strains normal to, and parallel to, the loading direction. Poisson s ratio of a RUCE for the loading direction-1 (ν 12U ) is obtained as ν 12U = ɛ 2 ɛ 1 (3) where ɛ 1 and ɛ 2 represent the strains of a RUCE in direction-1 and direction-2 respectively due to loading in direction-1. ɛ 1 can be obtained from equation (9). From 1(l), ɛ 2 can be expressed as 2δ cos θ ɛ 2 = h + 2l sin θ + 2s Thus the expression for Poisson s ratio of a RUCE for the loading direction-1 becomes 2 cos 2 θ ν 12U = ( 2 sin θ + 2 s + h l l ) sin θ (31) (32) Adhikari (Swansea) Mechanics of irregular honeycomb structures 28 June

37 Equivalent elastic properties of random irregular honeycombs Poisson s ratio ν 12 Poisson s ratio of the entire irregular honeycomb To derive the expression of equivalent Poisson s ratio for loading direction-1 of the entire irregular honeycomb structure (ν 12eq ), the Poisson s ratios for the constituting RUCEs (ν 12U ) are assembled. For obtaining ν 12eq, stress σ 1 is applied in direction-1. If the application of stress σ 1 in the j th strip is considered, total deformation of the j th strip in direction-1 is summation of individual deformations of the RUCEs in direction-1 of that particular strip. Thus from equation (12), using the basic definition of ν 12, ɛ 2 ˆν 12j L = m i=1 ɛ 2ij L ij ν U12ij (33) where ɛ 2 and ɛ 2ij are the strains in direction-2 of j th strip and individual RUCEs of j th strip respectively. ν U12ij represents the Poisson s ratio for loading direction-1 of a RUCE positioned at (i,j). ˆν 12j denotes the equivalent Poisson s ratio for loading direction-1 of the j th strip. Adhikari (Swansea) Mechanics of irregular honeycomb structures 28 June

38 Equivalent elastic properties of random irregular honeycombs Poisson s ratio ν 12 Poisson s ratio of the entire irregular honeycomb Ensuring the deformation compatibility condition ɛ 2 = ɛ 2ij for i = 1, 2,..., m in the j th strip, equation (33) leads to ˆν 12j = m L L ij (34) i=1 ν 12Uij Total deformation of the entire honeycomb structure in direction-2 under the application of stress σ 1 along the two opposite edges parallel to direction-2 is summation of the individual deformations in direction-2 of n number of strips. Thus n ɛ 2 B = ɛ 2j B j (35) j=1 Adhikari (Swansea) Mechanics of irregular honeycomb structures 28 June

39 Here ν 12Uij can be obtained from equation (32). Adhikari (Swansea) Mechanics of irregular honeycomb structures 28 June Equivalent elastic properties of random irregular honeycombs Poisson s ratio ν 12 Poisson s ratio of the entire irregular honeycomb Using the basic definition of ν 12 equation (35) becomes n ν 12eq ɛ 1 B = ν 12j ɛ 1j B j (36) where ν 12eq represents the equivalent Poisson s ratio for loading direction-1 of the entire irregular honeycomb structure. ɛ 1 and ɛ 1j denote the strain of entire honeycomb structure in direction-1 and strain of j th strip in direction-1 respectively. From the condition of deformation comparability ɛ 1 = ɛ 1j for j = 1, 2,..., n. Thus from equation (34) and equation (36): j=1 Equivalent ν 12 ν 12eq = 1 n B j j=1 m n l ij cos θ ij i=1 m l ij cos θ ij B j (37) j=1 ν 12Uij i=1

40 Equivalent elastic properties of random irregular honeycombs Poisson s ratio ν 21 Poisson s ratio ν 21 Poisson s ratio of a RUCE for the loading direction-2 (ν 21U ) is obtained as ν 21U = ɛ 1 ɛ 2 (38) where ɛ 1 and ɛ 2 represent the strains of a RUCE in direction-1 and direction-2 respectively due to loading in direction-2. ɛ 2 can be obtained from equation (19) and equation (2) as σ 2 l cos θ ɛ 2 = E s t 3 (h + 2s + 2l sin θ) ( ( cos 2l 3 cos 2 θ + 8s 3 2 ) ) α sin 3 α + cos2 β sin 3 + 2s 2 l(cot 2 α + cot 2 β) β We have with δ 1 = ( ) W 2 cos θ l 3 12E s I ɛ 1 = δ 1 sin θ l cos θ and W = 2σ 2 lb cos θ. (39) (4) Adhikari (Swansea) Mechanics of irregular honeycomb structures 28 June 216 4

41 Equivalent elastic properties of random irregular honeycombs Poisson s ratio ν 21 Poisson s ratio of a representative unit cell element (RUCE) Thus equation (4) reduces to ν 21U = ɛ 1 = σ 2l 3 sin θ cos θ E s t 3 (41) Thus the expression for Poisson s ratio of a RUCE for the loading direction-2 becomes ( h sin θ + 2 s ) + 2 sin θ l l ( s ) 3 ( cos 2 2 cos 2 α θ + 8 l sin 3 α + cos2 β sin 3 β ) ( s ) (cot 2 α + cot 2 β) l (42) Adhikari (Swansea) Mechanics of irregular honeycomb structures 28 June

42 Equivalent elastic properties of random irregular honeycombs Poisson s ratio ν 21 Poisson s ratio of the entire irregular honeycomb To derive the expression of equivalent Poisson s ratio for loading direction-2 of the entire irregular honeycomb structure (ν 21eq ), the Poisson s ratios for the constituting RUCEs (ν 21U ) are assembled. For obtaining ν 21eq, stress σ 2 is applied in direction-2. If the application of stress σ 2 in the j th strip is considered, total deformation of the j th strip in direction-1 is summation of individual deformations of the RUCEs in direction-1 of that particular strip. Thus, ɛ 1 L = Using the basic definition of ν 21 equation (43) leads to ˆν 21j ɛ 2 L = m ɛ 1ij L ij (43) i=1 m ν 21Uij ɛ 2ij L ij (44) where ˆν 21j represents the equivalent Poisson s ratio for loading direction-2 of the j th strip. i=1 Adhikari (Swansea) Mechanics of irregular honeycomb structures 28 June

43 Equivalent elastic properties of random irregular honeycombs Poisson s ratio ν 21 Poisson s ratio of the entire irregular honeycomb ɛ 2 and ɛ 2ij are the strains in direction-2 of j th strip and individual RUCEs of j th strip respectively. ν 21Uij represents the Poisson s ratio for loading direction-2 of a RUCE positioned at (i,j). To ensure the deformation compatibility condition ɛ 2 = ɛ 2ij for i = 1, 2,..., m in the j th strip. Thus equation (44) leads to ν 21j ˆ = m ν 21Uij l ij cos θ ij i=1 (45) m l ij cos θ ij i=1 Total deformation of the entire honeycomb structure in direction-2 under the application of stress σ 2 along the two opposite edges parallel to direction-1 is summation of the individual deformations in direction-2 of n number of strips. Thus n ɛ 2 B = ɛ 2j B j (46) j=1 Adhikari (Swansea) Mechanics of irregular honeycomb structures 28 June

44 Equivalent elastic properties of random irregular honeycombs Poisson s ratio ν 21 Poisson s ratio of the entire irregular honeycomb By definition of ν 21 equation (46) leads to ɛ 1 ν 21eq B = n j=1 ɛ 1j B j (47) ν 21j ˆ From the condition of deformation comparability ɛ 1 = ɛ 1j for j = 1, 2,..., n. Thus the equivalent Poisson s ratio for loading direction-2 of the entire irregular honeycomb structure: Equivalent ν 12 ν 21eq = n B j j=1 1 m l ij cos θ ij i=1 m ν 21Uij l ij cos θ ij i=1 n B j (48) j=1 Here ν 21Uij can be obtained from equation (42). Adhikari (Swansea) Mechanics of irregular honeycomb structures 28 June

45 Equivalent elastic properties of random irregular honeycombs Shear modulus (G 12 ) Shear modulus (G 12 ) To derive the expression of shear modulus (G 12U ) for a RUCE, shear stress τ is applied as shown below: (a) (b) (c) Figure: RUCE and free-body diagram used in the proposed analysis for G 12 Adhikari (Swansea) Mechanics of irregular honeycomb structures 28 June

46 Equivalent elastic properties of random irregular honeycombs Shear modulus (G 12 ) Elastic property of a representative unit cell element (RUCE) Total lateral movement of point D with respect to point H can be obtained as δ L = 2τl cos θ Et 3 ( (2ls 2 + h s3 sin α + 1 ) ) + (s + l sin θ)h 2 sin β (49) The shear strain γ for a RUCE can be expressed as δ L γ = 2s + h + 2l sin θ = 2τl cos θ Et 3 (2s + h + 2l sin θ) ( (2ls 2 + h s3 sin α + 1 ) ) (5) + (s + l sin θ)h 2 sin β Adhikari (Swansea) Mechanics of irregular honeycomb structures 28 June

47 Equivalent elastic properties of random irregular honeycombs Shear modulus (G 12 ) Elastic property of a representative unit cell element (RUCE) Thus the expression for shear modulus of a RUCE becomes G 12U = τ γ = Es ( t l ) 3 ( ( s ) 2 ( s 2 cos θ l l ( 2 s l ) + 2 sin θ + h l ) 3 ( 1 sinα + 1 ) + 1 sinβ 2 ( ) 3 h ( s + l l ) ( ) ) 2 h + sin θ l (51) Adhikari (Swansea) Mechanics of irregular honeycomb structures 28 June

48 Equivalent elastic properties of random irregular honeycombs Shear modulus (G 12 ) Elastic property of the entire irregular honeycomb To derive the expression of equivalent shear modulus of the entire irregular honeycomb structure (G 12eq ), the shear moduli for the constituting RUCEs (G 12U ) are assembled : (b) Idealized j th strip (a) Entire idealized irregular honeycomb structure (c) Idealized i th cell in j th strip Figure: Free-body diagrams of idealized irregular honeycomb structure in the proposed analysis of G 12 Adhikari (Swansea) Mechanics of irregular honeycomb structures 28 June

49 Equivalent elastic properties of random irregular honeycombs Shear modulus (G 12 ) Elastic property of the entire irregular honeycomb For obtaining G 12eq, shear stress τ is applied parallel to direction direction-1. If the equilibrium of forces for application of stress τ in the j th strip is considered: m τl = τ ij L ij (52) By definition of shear modulus equation (52) can be rewritten as i=1 Ĝ 12j γl = m G 12Uij γ ij L ij (53) i=1 where Ĝ12j represents the equivalent shear modulus of the j th strip. γ and γ ij are the shear strains of j th strip and individual RUCEs of the j th strip respectively. G 12Uij represents the shear modulus of a RUCE positioned at (i,j). Adhikari (Swansea) Mechanics of irregular honeycomb structures 28 June

50 Equivalent elastic properties of random irregular honeycombs Shear modulus (G 12 ) Elastic property of the entire irregular honeycomb To ensure the deformation compatibility condition γ = γ ij for i = 1, 2,..., m in the j th strip. Thus equation (53) leads to m G 12Uij l ij cos θ ij i=1 Ĝ 12j = (54) m l ij cos θ ij i=1 Total lateral deformation of one edge compared to the opposite edge of the entire honeycomb structure under the application of shear stress τ is the summation of the individual lateral deformations of n number of strips. Thus n γb = γ j B j (55) j=1 By definition of G 12 equation (55) leads to τ n τ j B = B G 12eq Gˆ j (56) 12j j=1 Adhikari (Swansea) Mechanics of irregular honeycomb structures 28 June 216 5

51 Equivalent elastic properties of random irregular honeycombs Shear modulus (G 12 ) Elastic property of the entire irregular honeycomb From equation (54) and (56), the equivalent shear modulus of the entire irregular honeycomb structure can be expressed as: Equivalent G 12 G 12eq = n B j j=1 1 m l ij cos θ ij i=1 m G 12Uij l ij cos θ ij i=1 n B j (57) j=1 Here G 12Uij can be obtained from equation (51). Adhikari (Swansea) Mechanics of irregular honeycomb structures 28 June

52 Uncertainty modelling and simulation Computational model and validation The analytical approach is capable of obtaining equivalent in-plane elastic properties for irregular honeycombs from known random spatial variation of cell angle and material properties of the honeycomb cells. The homogenised properties depend on the ratios h/l, t/l, s/l and the angles θ, α, β. In addition, the two Young s moduli and shear modulus also depend on E s. We show results for h/l = 1.5 and three values of cell angle θ, namely: 3, 45 and 6. As the two Young s moduli and shear modulus of low density honeycomb are proportional to E s ρ 3 (Zhu et al., 21), the non-dimensional results for elastic moduli E 1, E 2, ν 12, ν 21 and G 12 have been obtained using Ē 1 = E 1eq E s ρ 3, Ē 2 = E 2eq E s ρ 3, ν 12 = ν 12eq, ν 21 = ν 21eq and G 12 = G 12eq E s ρ 3 respectively, where. represents the non-dimensional elastic modulus and ρ is the relative density of honeycomb (ratio of the planar area of solid to the total planar area of the honeycomb). Adhikari (Swansea) Mechanics of irregular honeycomb structures 28 June

53 Uncertainty modelling and simulation Computational model and validation (a) θ = 6 (b) θ = 45 Figure: Regular honeycomb with different θ values Adhikari (Swansea) Mechanics of irregular honeycomb structures 28 June

54 Uncertainty modelling and simulation Computational model and validation For the purpose of finding the range of variation in elastic moduli due to spatial uncertainty, cell angles and material properties can be perturbed following a random distribution within specific bounds. We show results for spatial irregularity in the cell angles only Frequency 1 Frequency Cell angle (a) Distribution of cell angle (θ) Inclination angle α (b) Distribution of the inclination angle (α) Figure: Typical statistical distribution of cell angle (θ) and inclination angle α (number of RUCE: 1681) Adhikari (Swansea) Mechanics of irregular honeycomb structures 28 June

55 Uncertainty modelling and simulation Computational model and validation The maximum, minimum and mean values of non-dimensional in-plane elastic moduli for different degree of spatially random variations in cell angles ( θ =, 1, 3, 5, 7 ) are calculated using both direct finite element simulation and the derived closed-form expressions. For a particular cell angle θ, results have been obtained using a set of uniformly distributed 1 random samples in the range of [θ θ, θ + θ]. The set of input parameter for a particular sample consists of N number of cell angles in the specified range, where N(= n m) is the total number of RUCEs in the entire irregular honeycomb structure. We used 1681 RUCEs (as this was needed for convergence of the deterministic case). The quantities h and θ have been considered as the two random input parameters while α, β and l are dependent features. Adhikari (Swansea) Mechanics of irregular honeycomb structures 28 June

56 Results and discussions Numerical results for the homogenised in-plane properties Longitudinal elastic modulus (E 1 ) θ = 3 o θ = 45 o θ = 6 o Ē Random variation in cell angle (degree) h/l = 1.5, Ē 1 = E 1eq /E sρ Adhikari (Swansea) Mechanics of irregular honeycomb structures 28 June

57 Results and discussions Numerical results for the homogenised in-plane properties Transverse elastic modulus (E 2 ) θ = 3 o θ = 45 o θ = 6 o Ē Random variation in cell angle (degree) h/l = 1.5, Ē 2 = E 2eq /E sρ 3 Adhikari (Swansea) Mechanics of irregular honeycomb structures 28 June

58 Results and discussions Numerical results for the homogenised in-plane properties Poisson s ratio ν θ = 3 o θ = 45 o θ = 6 o ν Random variation in cell angle (degree) h/l = 1.5 Adhikari (Swansea) Mechanics of irregular honeycomb structures 28 June

59 Results and discussions Numerical results for the homogenised in-plane properties Poisson s ratio ν θ = 3 o θ = 45 o θ = 6 o ν Random variation in cell angle (degree) h/l = 1.5 Adhikari (Swansea) Mechanics of irregular honeycomb structures 28 June

60 Results and discussions Numerical results for the homogenised in-plane properties Shear modulus (G 12 ) G θ = 3 o θ = 45 o θ = 6 o Random variation in cell angle (degree) h/l = 1.5, G 12 = G 12eq /E sρ 3 Adhikari (Swansea) Mechanics of irregular honeycomb structures 28 June 216 6

61 Results and discussions Numerical results for the homogenised in-plane properties Probability density function of E 2 Probability density function FEM Analytical Ē 2 h/l = 1.5, Ē 2 = E 2eq /E regular, θ = 45, θ = 5 Adhikari (Swansea) Mechanics of irregular honeycomb structures 28 June

62 Results and discussions Numerical results for the homogenised in-plane properties Ergodic behaviour of E 2 : spread of values h/l = 1.5, Ē 2 = E 2eq /E regular, θ = 45, θ = 5 Adhikari (Swansea) Mechanics of irregular honeycomb structures 28 June

63 Results and discussions Numerical results for the homogenised in-plane properties Ergodic behaviour of E 2 : coefficient of variation h/l = 1.5, θ = 45, θ = 5 Adhikari (Swansea) Mechanics of irregular honeycomb structures 28 June

64 Results and discussions Numerical results for the homogenised in-plane properties Ergodic behaviour of ν 21 : coefficient of variation h/l = 1.5, θ = 45, θ = 5 Adhikari (Swansea) Mechanics of irregular honeycomb structures 28 June

65 Results and discussions Numerical results for the homogenised in-plane properties Ergodic behaviour of G 12 : coefficient of variation h/l = 1.5, θ = 45, θ = 5 Adhikari (Swansea) Mechanics of irregular honeycomb structures 28 June

66 Results and discussions Main observations Main observations The elastic moduli obtained using the analytical method and by finite element simulation are in good agreement - establishing the validity of the closed-form expressions. The number of input random variables (cell angle) increase with the number of cells. The variation in E 1 and ν 12 due to spatially random variations in cell angles is very less, while there is considerable amount of reductions in the values of E 2, ν 21 and G 12 with increasing degree of irregularity. Longitudinal Young s modulus, transverse Young s modulus and shear modulus are functions of both structural geometry and material properties of the irregular honeycomb (i.e. ratios h/l, t/l, s/l and angles θ, α, β and E s ), while the Poisson s ratios depend only on structural geometry of irregular honeycombs (i.e. ratios h/l, t/l, s/l and angles θ, α, β) For large number of random cells ( 17), we observe the emergence of an effective ergodic behaviour - ensemble statistics become close to single sample statistics. Adhikari (Swansea) Mechanics of irregular honeycomb structures 28 June

67 Conclusions Conclusions Adhikari (Swansea) Mechanics of irregular honeycomb structures 28 June

68 Conclusions Some publications Mukhopadhyay, T. and Adhikari, S., Free vibration of sandwich panels with randomly irregular honeycomb core, ASCE Journal of Engineering Mechanics, in press. Mukhopadhyay, T. and Adhikari, S., Equivalent in-plane elastic properties of irregular honeycombs: An analytical approach, International Journal of Solids and Structures, 91[8] (216), pp Mukhopadhyay, T. and Adhikari, S., Effective in-plane elastic properties of auxetic honeycombs with spatial irregularity, Mechanics of Materials, 95[2] (216), pp Adhikari (Swansea) Mechanics of irregular honeycomb structures 28 June

69 References El-Sayed, F. K. A., Jones, R., Burgess, I. W., A theoretical approach to the deformation of honeycomb based composite materials. Composites 1 (4), Gibson, L., Ashby, M. F., Cellular Solids Structure and Properties. Cambridge University Press, Cambridge, UK. Goswami, S., 26. On the prediction of effective material properties of cellular hexagonal honeycomb core. Journal of Reinforced Plastics and Composites 25 (4), Li, K., Gao, X. L., Subhash, G., 25. Effects of cell shape and cell wall thickness variations on the elastic properties of two-dimensional cellular solids. International Journal of Solids and Structures 42 (5-6), Masters, I. G., Evans, K. E., Models for the elastic deformation of honeycombs. Composite Structures 35 (4), Zhang, J., Ashby, M. F., The out-of-plane properties of honeycombs. International Journal of Mechanical Sciences 34 (6), Zhu, H. X., Hobdell, J. R., Miller, W., Windle, A. H., 21. Effects of cell irregularity on the elastic properties of 2d voronoi honeycombs. Journal of the Mechanics and Physics of Solids 49 (4), Zhu, H. X., Thorpe, S. M., Windle, A. H., 26. The effect of cell irregularity on the high strain compression of 2d voronoi honeycombs. International Journal of Solids and Structures 43 (5), Adhikari (Swansea) Mechanics of irregular honeycomb structures 28 June

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